We start by proving existence of solutions to the time-differentiated version of our original problem (REF ). We thus consider
\( \left\lbrace \begin{aligned}\alpha p_{tt} - b p̥_t -c^2p̥ =&-\alpha _t p_t + \sigma \nabla q\cdot \nabla p+\sigma \nabla \phi \cdot \nabla p_t \hspace*{-11.38109pt} &&\ \text{ in }\Omega \times (0,T), \\[1mm]p=&\,0 &&\ \text{ on } \partial \Omega \times (0,T),\\[1mm](p, p_t)=&\,(\psi _1, \psi _2) &&\ \mbox{ in }\Omega \times \lbrace 0\rbrace ,\end{aligned} \right.\) 
where the function \(\psi _2\)  is defined by (REF ).
The proof can be carried out as before via smooth Faedo–Galerkin approximations in space, by projecting the problem (REF ) onto the span \(V_n\)  of the first \(n\)  eigenfunctions of the Laplacian pointwise in time.
We focus here on deriving the crucial uniform bound for the approximate solution since the remaining steps are analogous to the proof of Proposition REF . For simplicity of notation, we drop the superscript \(n\)  in the approximate solution.
In the first step, we multiply (REF ) by \(2 p_t\)  and integrate over \(\Omega \) :
\((\alpha p_{tt} - b p̥_t -c^2p̥, 2 p_t)_{L^2}=(-\alpha _t p_t + \sigma \nabla q\cdot \nabla p + \sigma \nabla \phi \cdot \nabla p_t, 2 p_t)_{L^2}.\) 
When deriving the energy bound, we can rely on the identity
\(\begin{aligned}(\alpha p_{tt}, 2 p_t)_{L^2}=&\,\, (\alpha p_{tt}], p̥_t)_{L^2} \\=&\, \frac{\textup {d}}{\textup {d}t}\frac{1}{2} (\alpha p̥_t, p̥_t)_{L^2} +\frac{1}{2} \kappa (q_t p̥_t, p̥_t)_{L^2}-(\kappa q̥\, p_{tt} +\kappa \nabla q\cdot \nabla p_{tt}, p̥_t)_{L^2}\\=&\, \frac{\textup {d}}{\textup {d}t}\frac{1}{2} (\alpha p̥_t, p̥_t)_{L^2} + (f_1(\phi ,q,p), p̥_t)_{L^2},\end{aligned}\) 
where we have used the fact that \(p̥\)  vanishes on the boundary for smooth approximations of the problem and introduced the short-hand notation
\(\begin{aligned}f_1(\phi ,q,p) = \frac{1}{2} \kappa q_t -\kappa q̥ p_{tt}- \kappa \nabla q \cdot \nabla p_{tt}.\end{aligned}\) 
Recalling that \(p_{tt}==p̥=0\)  on \(\partial \Omega \) , which via the PDE implies that also \(\nabla q \cdot \nabla p+\nabla \phi \cdot \nabla p_t\)  vanishes on the boundary, we further find that
\(\begin{aligned}&(-\alpha _t p_t + \sigma \nabla q\cdot \nabla p + \sigma \nabla \phi \cdot \nabla p_t, 2 p_t)_{L^2}\\=&\,\begin{multlined}[t](-\alpha _t p_t], )_{L^2}+\sigma (\nabla q\cdot \nabla p + \nabla \phi \cdot \nabla p_t], )_{L^2}.\end{multlined}\end{aligned}\) 
Furthermore, we have
\(\begin{aligned}&-\alpha _t p_t + \sigma \nabla q\cdot \nabla p + \sigma \nabla \phi \cdot \nabla p_t]\\=&\,\begin{multlined}[t] \kappa \Delta q_t \, p_t + 2\kappa \nabla q_t\cdot \nabla p_t + \kappa q_t\,p̥_t+ \sigma \nabla q̥\cdot \nabla p+ 2\sigma D^2 q:D^2 p\\+ \sigma \nabla q\cdot \nabla p̥+ \sigma \nabla \cdot \nabla p_t+ 2\sigma D^2 \phi :D^2 p_t+ \sigma \nabla \phi \cdot \nabla p̥_t \end{multlined}\\=&\, f_2(\phi ,q,p)+\sigma \nabla \phi \cdot \nabla p̥_t,\end{aligned}\) 
where we have introduced another short-hand notation
\(\begin{aligned}f_2(\phi ,q,p)= \kappa \Delta q_t \, p_t + 2\kappa \nabla q_t\cdot \nabla p_t + \kappa q_t\,p̥_t+ \sigma \nabla q̥\cdot \nabla p+ 2\sigma D^2 q:D^2 p\\+ \sigma \nabla q\cdot \nabla p̥+ \sigma \nabla \cdot \nabla p_t+ 2\sigma D^2 \phi :D^2 p_t\,.\end{aligned}\) 
By elliptic regularity the Hessian \(D^2 v=(\partial _{x_i} \partial _{x_j} v)_{i,j}\)  satisfies
\(\Vert D^2 v\Vert _{L^p}\le C_{\textup {H}} \Vert \Delta v\Vert _{L^p}\quad \mbox{ for all }v\in H_0^1(\Omega )\cap W^{2,p}(\Omega )\, \quad p\in (1,6],\) 
see, e.g., [1]}.

Due to the PDE, we know that
\(\begin{aligned}p_{tt}=&\,\alpha ^{-1}\left(b p̥_t +c^2p̥ + \kappa q_t p_t + \sigma \nabla q\cdot \nabla p + \sigma \nabla \phi \cdot \nabla p_t\right),\\\nabla p_{tt}=&\, \begin{multlined}[t]\kappa \alpha ^{-2}\nabla q\,\left(b p̥_t +c^2p̥ + \kappa q_t p_t + \sigma \nabla q\cdot \nabla p + \sigma \nabla \phi \cdot \nabla p_t\right)\\+\alpha ^{-1}\left(b \nabla p̥_t +c^2\nabla p̥ + \kappa \nabla q_t p_t+ \kappa q_t \nabla p_t+ \sigma D^2 q\,\nabla p \right. \\ \left.+ \sigma D^2 p\,\nabla q + \sigma D^2\phi \,\nabla p_t + \sigma D^2 p_t\,\nabla \phi \right). \end{multlined}\end{aligned}\) 
Altogether, we have the energy identity
\(\begin{aligned}&\frac{1}{2}\frac{\textup {d}}{\textup {d}t}\Vert \sqrt{\alpha }p̥_t\Vert _{L^2}^2 + b \Vert \nabla \Delta p_t\Vert _{L^2}^2 + c^2 \frac{1}{2}\frac{\textup {d}}{\textup {d}t} \Vert \nabla p̥\Vert _{L^2}^2 \\[1mm]&= \begin{multlined}[t](-f_1(\phi ,q,p)+f_2(\phi ,q,p)+\sigma \nabla \phi \cdot \nabla p̥_t, p̥_t)_{L^2}.\end{multlined}\end{aligned}\) 
We can estimate the right-hand side by employing the following bound:
\(\begin{aligned}\Vert f_1(\phi ,q,p)\Vert _{L^2}\le &\, \begin{multlined}[t] \frac{1}{2}|\kappa | C_{H^2, L^\infty }\Vert q̥_t\Vert _{L^2}\Vert p̥_t\Vert _{L^2}\\+ (|{\kappa }|C_{H^2, L^\infty }+|\kappa |C_{H^1, L^4}^2) \Vert \nabla q̥\Vert _{L^2} \Vert \nabla p_{tt}\Vert _{L^2}. \end{multlined}\end{aligned}\) 
Furthermore, by virtue of identities (REF ), we have
\(\begin{aligned}\Vert \nabla p_{tt}\Vert _{L^2}\le &\, \begin{multlined}[t]\frac{|\kappa |}{\underline{\alpha }^2} C_{H^2, L^\infty }\Vert \nabla q̥\Vert _{L^2}\left(b\Vert p̥_t\Vert _{L^2}+c^2\Vert p̥\Vert _{L^2}+C_{H^1, L^4}^2 \left(|\kappa |\Vert \nabla q_t\Vert _{L^2}\Vert \nabla p_t\Vert _{L^2} \right. \right.\\ \left. \left.+ |\sigma |\Vert q̥\Vert _{L^2}\Vert p̥\Vert _{L^2} + |\sigma |\Vert \Vert _{L^2}\Vert p̥_t\Vert _{L^2}\right) \vphantom{C_{H^2, L^\infty }}\right)\\+\frac{1}{\underline{\alpha }} \left(b \Vert \nabla p̥_t\Vert _{L^2}+c^2\Vert \nabla p̥\Vert _{L^2}+2|\kappa | C_{H^1, L^4}C_{\textup {PF}}^{1/2} \Vert q̥_t\Vert _{L^2}\Vert p̥_t\Vert _{L^2}\right. \\ \left.+|\sigma |C_{\textup {H}} (C_{H^2, L^\infty }+C_{H^1, L^4}^2) (\Vert \nabla q̥\Vert _{L^2}\Vert p̥\Vert _{L^2}+\Vert \nabla \Vert _{L^2}\Vert p̥_t\Vert _{L^2})\right). \end{multlined}\end{aligned}\) 
We note that also
\(\begin{aligned}\Vert f_2(\phi ,q,p)\Vert _{L^2}\le &\, \begin{multlined}[t]2|\kappa |(C_{H^2, L^\infty }+C_{H^1, L^4}^2) \Vert q̥_t\Vert _{L^2}\Vert p̥_t\Vert _{L^2} \\+ 2|\sigma |(C_{H^2, L^\infty }+C_{\textup {H}}^2 C_{H^1, L^4}^2) \Vert \nabla q̥\Vert _{L^2}\Vert \nabla p̥\Vert _{L^2} \\+ |\sigma |(C_{\textup {H}} C_{H^2, L^\infty }+2 C_{H^1, L^4}^2) \Vert 2\phi \Vert _{L^2}\Vert p̥_t\Vert _{L^2} \end{multlined}\end{aligned}\) 
and
\(\begin{aligned}2 (\nabla \phi \cdot \nabla p̥_t, p̥_t)_{L^2} = (\nabla \phi , \nabla (p̥_t)^2)_{L^2} =&\, (, (p̥_t)^2)_{L^2}\\\le &\, C_{H^2, L^\infty }\Vert \Delta ^2\phi \Vert _{L^2}\Vert p̥_t\Vert _{L^2}^2.\end{aligned}\) 
Under the regularity assumptions (REF ), similarly to the proof of Proposition REF , by using Young's and Gronwall's inequalities, we arrive at an energy estimate of the form
\( \begin{aligned}& \begin{multlined}[t] \Vert \nabla p_{tt}\Vert _{L^2 (L^2)}^2\,+\sup _{t \in (0,T)} \Vert (t)\Vert _{L^2}^2\,+\sup _{t \in (0,T)} \Vert \nabla p̥(t)\Vert _{L^2}^2+b\Vert \nabla \Vert _{L^2 (L^2)}^2\end{multlined}\\\lesssim &\, \begin{multlined}[t] \Vert \nabla _1\Vert _{L^2}^2+\Vert _2\Vert _{L^2}^2,\end{multlined}\end{aligned}\) 
first of all, for the Galerkin approximations. Via weak limits we obtain existence, uniqueness, and the energy estimate (REF ) also for a solution to (REF ).
Integrating with respect to time and using compatibility of the initial data (REF ) yields a unique solution to (REF ).
Additional regularity
The terms containing \(\Vert \Delta ^2\phi \Vert _{L^2}\Vert p̥_t\Vert _{L^2}\)  above and in \(f_2\)  show that another energy estimate is needed to obtain an overall bound that is independent of \(b\)  for the nonlinear problem. To this end, we exploit elliptic regularity and the fact that by the PDE, \(\)  satisfies
\((b\partial _t+c^2)= r \mbox{ with } r=\alpha p_t -\sigma \nabla \phi \cdot \nabla p.\) 
Thus in case \(b>0\) , we have
\((t) = g_0(t) (0) + \int _0^t g^{\prime }(t-s) r(s)\, \textup {d} s \) 
with
\(g(t)=-c^{-2} \exp {\left(-\frac{c^2}{b} t\right)},\qquad g_0(t)=\exp {\left(-\frac{c^2}{b} t\right)}\in [0,1].\) 
By positivity of \(g^{\prime }\) , we have
\(\Vert g^{\prime }\Vert _{L^1(0,t)}=\int _0^t g^{\prime }(s)\, \textup {d} s = g(t)-g(0) = c^{-2}\left(1-\exp {\left(-\frac{c^2}{b} t\right)}\right)\le c^2.\) 
Therefore,
\(\Vert 2 \psi (t)\Vert _{L^2}\le \Vert 2 \psi (0)\Vert _{L^2} + c^{-2} \Vert r̥\Vert _{L^\infty (0,t;L^2(\Omega )}.\) 
In case \(b=0\) , estimate (REF ) (without the \(\Vert 2 \psi (0)\Vert _{L^2}\)  term) immediately follows from (REF ). The right hand-side norm can be estimated as follows:
\(\begin{aligned}&\Vert r̥\Vert _{L^\infty (L^2)}\\=&\, \Vert \kappa q̥ \, p_t + 2\kappa \nabla q\cdot \nabla p_t + \alpha \,p̥_t+ \sigma \nabla \cdot \nabla p+ 2\sigma D^2 \phi :D^2 p+ \sigma \nabla \phi \cdot \nabla p̥\Vert _{L^\infty (L^2)}\\\le &\, \begin{multlined}[t]{(1+|\kappa | (2C_{H^2, L^\infty }+2C_{\textup {H}}^2 C_{H^1, L^4}^2)}\Vert q̥\Vert _{L^\infty (L^2)} )\Vert p̥_t\Vert _{L^\infty (L^2)} \\+2 |\sigma | ({C_{\textup {H}}}C_{H^2, L^\infty }+C_{\textup {H}}^2 C_{H^1, L^4}^2)\Vert \nabla \Vert _{L^\infty (L^2)} \Vert \nabla p̥\Vert _{L^\infty (L^2)}\,,\end{multlined}\end{aligned}\) 
and therefore bounded by means of the already established estimate (REF ).

Here we have used the elliptic regularity result (see, e.g., [2]})
\(\Vert v\Vert _{H^3}\le C_{\textup {H}} \Vert (-\Delta )^{3/2} v\Vert _{L^2}\quad \mbox{ for all }v\in H_0^1(\Omega )\cap H^3(\Omega ),\) 
and estimated
\(\Vert \nabla p\Vert _{L^\infty }\le C_{H^2, L^\infty } \Vert p\Vert _{H^3}\le C_{H^2, L^\infty } C_{\textup {H}} \Vert (-\Delta )^{3/2} p\Vert _{L^2}= C_{H^2, L^\infty } C_{\textup {H}} \Vert \nabla \Delta p\Vert _{L^2}\,,\) 
and in the same manner bounded \(\Vert \nabla \phi \Vert _{L^\infty }\) , due to the fact that \(\Delta p\)  and \(\Delta \phi \)  vanish on \(\partial \Omega \) .

Furthermore, by (REF ), the function \(\psi \)  satisfies the boundary conditions
\(\begin{aligned}&\psi \vert _{\partial \Omega }=0\,, \\&\vert _{\partial \Omega }={\mathrm {bdy}}:={\left\lbrace \begin{array}{ll}g_0(t) \vert _{\partial \Omega }(0) + \int _0^t g^{\prime }(t-s)(\sigma \nabla \phi \cdot \nabla p)\vert _{\partial \Omega }(s)\, \textup {d} s &\mbox{ if }b>0\\(\sigma \nabla \phi \cdot \nabla p)\vert _{\partial \Omega }&\mbox{ if }b=0\,,\end{array}\right.}\end{aligned}\) 
where
\(\begin{aligned}\Vert (\sigma \nabla \phi \cdot \nabla p)\vert _{\partial \Omega }\Vert _{L^\infty (0,T;H^{3/2}(\partial \Omega ))}\le C_{\textup {tr}} \Vert \sigma \nabla \phi \cdot \nabla p\Vert _{L^\infty (0,T;{H^2(\Omega )})}\end{aligned}\) 
can again be estimated by means of (REF ).
Thus we can invoke higher elliptic regularity [3]} to first conclude
\(\Vert \psi \Vert _{L^\infty (0,T;H^4(\Omega ))}\le C_{\textup {H}}\Bigl ( \Vert \Vert _{L^\infty (0,T;H^2(\Omega )}+\Vert \psi \Vert _{L^\infty (L^2)}\Bigr )\) 
and then further estimate
\(\Vert \Vert _{L^\infty (0,T;H^2(\Omega ))}\le C_{\textup {H}}\Bigl ( \Vert r̥\Vert _{L^\infty (L^2)} + {\Vert \mathrm {bdy}\Vert _{L^\infty (H^{3/2}(\partial \Omega ))}} + \Vert \psi \Vert _{L^\infty (L^2)}\Bigr ).\) 
Altogether, by taking into account the derived bounds, we obtain the energy estimate (REF ); cf. (REF ), where we estimate \(\Vert _2\Vert _{L^2}^2\)  by means of (REF ).

Uniform bounds for the Kuznetsov equation
We next employ a fixed-point argument to analyze the Kuznetsov equation for small data as well as obtain a uniform in \(b\)  energy bound for its solution. To this end, we introduce the mapping
\(\mathcal {T}_{\textup {K}}:\phi \mapsto \psi ,\) 
where, similarly to before, \(\phi \)  will belong to a suitably chosen ball in the space \(X^{\textup {K}}\) , and \(\psi \)  will solve the linearized Kuznetsov equation
\( \alpha \psi _{tt} -c^2- b _t =\, \sigma \nabla \phi \cdot \nabla \psi _t\) 
with \(\alpha =1-\kappa \phi _t\)  and initial data \((\psi (0), \psi _t(0))=(\phi (0), \phi _t(0))=(\psi _0, \psi _1)\) . We note that the mapping \(\mathcal {T}_{\textup {K}}\)  is well-defined on account of Proposition REF , whose assumptions we verify below.
Theorem 6.1 
Let \(\Omega \subset \mathbb {R}^n\) , where \(n \in \lbrace 1, 2, 3\rbrace \) , be bounded and \(C^4\)  regular. Let \(\kappa \) , \(\sigma \in \mathbb {R}\) . Furthermore, let \(b \in [0, \bar{b})\)  and let the initial data \((\psi _0, \psi _1) \)  satisfy the regularity assumption (REF ) with
\(\Vert \psi _0\Vert ^2_{H^4}+\Vert \psi _1\Vert ^2_{H^3} \le \delta ^2.\) 
There exist \(\tilde{\delta }>0\)  and a final time \(T=T(\delta )>0\) , independent of \(b\) , such that if
\({\Vert \psi _0\Vert ^2_{H^2}+\Vert \psi _1\Vert ^2_{H^2} \le \tilde{\delta }^2,}\) 
then there exists a unique solution \(\psi \in X^{\textup {K}}\)  of the problem
\( \left\lbrace \begin{aligned}\psi _{tt}- b _t - c^2=&\, \frac{1}{2} (\kappa \psi _t^2 + \sigma |\nabla \psi |^2)_{t} &&\quad \text{ in }\Omega \times (0,T), \\[1mm]\psi =&\,0 &&\quad \text{ on } \partial \Omega \times (0,T),\\[1mm](\psi , \psi _t)=&\,(\psi _0, \psi _1) &&\quad \mbox{ in }\Omega \times \lbrace 0\rbrace ,\end{aligned} \right.\) 
where \(X^{\textup {K}}\)  is defined in (REF ). Furthermore, the solution \(\psi \)  satisfies the estimate (REF ).

We begin by proving that the mapping \(\mathcal {T}_{\textup {K}}:\phi \mapsto \psi \)  is well-defined and a self-mapping on
\(\begin{aligned}M_{\textup {K}} = \left\lbrace  \vphantom{\sup _{t \in (0,T)} \Vert 2 \psi (t)\Vert _{L^2}^2\le R_{\textup {K}}^2 }\right. \phi \in X^{\textup {K}}\, : \, &\phi (0)= \psi _0, \ \phi _t(0)=\psi _1,\\&\ \left. \vphantom{X^{\textup {K}}}\Vert \phi _t\Vert _{L^\infty (L^\infty )} \le \frac{1}{2 |\kappa |}, \right.\ \left.\Vert \phi \Vert _{X^{\textup {K}}} \le R_{\textup {K}}\ \vphantom{\sup _{t \in (0,T)} \Vert 2 \psi (t)\Vert _{L^2}^2\le R_{\textup {K}}^2 } \right\rbrace ,\end{aligned}\) 
where the radius \(R_{\textup {K}}\)  will be specified below. We first note that the conditions of Proposition REF  are satisfied. Indeed, regularity assumptions (REF ) and (REF ) hold with the uniform bounds
\(\begin{aligned}\Vert \phi \Vert _{L^\infty (H_\diamondsuit ^4)} \le \, R_{\textup {K}}, \qquad \Vert \alpha \Vert _{X^{\textup {K}}_{\alpha }} \lesssim \, 1+ R_{\textup {K}}.\end{aligned}\) 
Furthermore, we have
\(\begin{aligned}\Vert 1-\alpha \Vert _{L^\infty (L^\infty )}\le |\kappa |\Vert \phi _t\Vert _{L^\infty (L^\infty )} \le \frac{1}{2}.\end{aligned}\) 
We can thus take \(\underline{\alpha }=\frac{1}{2}\)  and \(\overline{\alpha }=\frac{3}{2}\)  in (REF ), and so the non-degeneracy condition is fulfilled as well. Then Proposition REF  applied to \(\psi =\mathcal {T}_{\textup {K}}(\phi )\)  implies that the mapping is well-defined and, moreover,
\( \begin{aligned}\Vert \psi \Vert ^2_{X^{\textup {K}}} \le \, C_{\textup {lin}, \textup {K}}(T, R_{\textup {K}})(\Vert 2 \psi _0\Vert _{L^2}^2+\Vert \nabla _1\Vert _{L^2}^2+\Vert \psi _0\Vert _{L^2}^2),\end{aligned}\) 
where \(C_{\textup {lin}, \textup {K}}(T, R_{\textup {K}})\)  is determined by the hidden constant in the linear bound (REF )

and by inspection of (REF ) can be seen to have a form similar to the Westervelt case (REF )
\(C_{\textup {lin}}(T,R)= C_1 \exp {\left(C_2(|\kappa |+|\sigma |)(R_{\textup {K}}+R_{\textup {K}}^2+R_{\textup {K}}^3)T\right)},\) 
with constants \(C_1\)  and \(C_2\)  independent of \(R_{\textup {K}}\) , \(T\) , and \(b \in [0, \bar{b})\)  for fixed \(\bar{b}\) .

Therefore, provided that \(R_{\textup {K}}\)  and final time \(T\)  are chosen so that
\(C_{\textup {lin}, \textup {K}}(T,R_{\textup {K}}) (\Vert 2 \psi _0\Vert _{L^2}^2+\Vert \nabla _1\Vert _{L^2}^2+\Vert \psi _0\Vert _{L^2}^2) \le R_{\textup {K}}^2,\) 
the solution \(\psi \)  satisfies the \(R_{\textup {K}}^2\)  bound in (REF ).
To obtain the \(1/{(2|\kappa |)}\)  bound, we will, similarly to the Westervelt case, derive a uniform in \(b\)  estimate of \(\Vert \psi _t \Vert _{L^\infty (H^1)}\)  and then rely on Agmon's interpolation inequality. By multiplying the time-differentiated equation (REF ) with \(p_{t}\)  and integrating over space and time, we arive at
\( \begin{aligned}&\frac{1}{2} \left\lbrace \Vert \sqrt{\alpha (t)}p_t(t)\Vert ^2_{L^2}+c^2\Vert \nabla p(t)\Vert ^2_{L^2} \right\rbrace \Big \vert _0^t+b \int _0^t\Vert \nabla p_t\Vert ^2\, \textup {d}x\textup {d}s\\=&-\frac{1}{2} \int _0^t(\alpha _t p_t, p_t)_{L^2}\, \textup {d} s +\sigma \int _0^t (\nabla q \cdot \nabla p+\nabla \phi \cdot \nabla p_t, p_t)\, \textup {d} s \\\lesssim &\,\begin{multlined}[t] \Vert \alpha _t\Vert _{L^\infty (L^\infty )}\Vert p_t\Vert ^2_{L^2(0,t;L^2)}+\Vert \nabla q\Vert _{L^\infty (L^\infty )}\Vert \nabla p\Vert _{L^2(0, t;L^2)}\Vert p_t\Vert _{L^2(0, t;L^2)}\\+\sigma \int _0^t (\nabla \phi \cdot \nabla p_t, p_t)\, \textup {d} s . \end{multlined}\end{aligned}\) 
To estimate the last term on the right, we can first rely on the identity
\( \begin{aligned}& \int _0^t \int _{\Omega } (\nabla \phi \cdot \nabla p_t) p_t \, \textup {d}x\textup {d}s\\=&\, \int _0^t \int _{\Omega } (- ) p_t p_t \, \textup {d}x\textup {d}s- \int _0^t \int _{\Omega } p_t (\nabla \phi \cdot \nabla p_t) \, \textup {d}x\textup {d}s\end{aligned}\) 
since \(p_t=0\)  on \(\partial \Omega \) , which then leads to the bound
\( \begin{aligned}\left|2\int _0^t \int _{\Omega } (\nabla \phi \cdot \nabla \overline{\psi }_t) p_t \, \textup {d}x\textup {d}s\right|=&\, \left| \int _0^t \int _{\Omega } (- ) \overline{\psi }_t p_t \, \textup {d}x\textup {d}s\right|\\\le &\, \Vert \Vert _{L^\infty (0,t; L^\infty )}\Vert p_t\Vert _{L^2(0,t; L^2)}^2.\end{aligned}\) 
Thus, utilizing the above estimate in (REF ) and employing Gronwall's inequality leads to
\(\begin{aligned}&\Vert p_t(t)\Vert ^2_{L^2}+\Vert \nabla p(t)\Vert ^2_{L^2} \\\le &\, \tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_1\exp (\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_2 R_{\textup {K}}T)(\Vert \psi _2\Vert ^2_{L^2}+\Vert \nabla \psi _1\Vert ^2_{L^2} )\\\le &\,\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_1\exp (\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_2 R_{\textup {K}}T)(\Vert \alpha (0)^{-1} (b _1 +c^2_0 + \sigma \nabla \psi _0\cdot \nabla \psi _1)\Vert ^2_{L^2}+\Vert \nabla \psi _1\Vert ^2_{L^2} ),\end{aligned}\) 
where we have also relied on the compatibility of initial data (REF ) in the last line. From here, noting that
\(\Vert \nabla \psi _0\cdot \nabla \psi _1\Vert _{L^2} \le \Vert \nabla \psi _0\Vert _{L^4}\Vert \nabla \psi _1\Vert _{L^4} \lesssim \Vert \nabla \psi _0\Vert _{H^1}\Vert \nabla \psi _1\Vert _{H^1},\) 
we conclude that there exist positive constants \(\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_2\)  and \(\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_3\) , independent of \(b\) , such that
\(\begin{aligned}\Vert p_t(t)\Vert ^2_{L^2}+\Vert \nabla p(t)\Vert ^2_{L^2}\le \,\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_3\exp (\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_2 R_{\textup {K}}T)(\Vert _1\Vert ^2_{L^2} +\Vert _0\Vert ^2_{L^2}+\Vert \nabla \psi _1\Vert ^2_{L^2} )\end{aligned}\) 
for all \(t \in [0,T]\) . The non-degeneracy bound follows from the above estimate and Agmon's inequality:
\(\Vert p\Vert _{L^\infty (L^\infty )} \le C_{\textup {A}}\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_4\exp (\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_5 RT)(\Vert _1\Vert ^2_{L^2} +\Vert _0\Vert ^2_{L^2}+\Vert \nabla \psi _1\Vert ^2_{L^2} )^{1/4}R_{\textup {K}}^{1/2},\) 
for some positive constants \(\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_4\) , \(\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_5\)  independent of \(b\) , if we choose \(\tilde{\delta }>0\)  small enough, so that
\(C_{\textup {A}}\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_4\exp (\tilde{\raisebox {0pt}[0.85]{\tilde{C}}}_5 RT)\sqrt{\tilde{\delta }}\,R_{\textup {K}}^{1/2} \le \frac{1}{2 |\kappa |}.\) 
Therefore,
\(\mathcal {T}_{\textup {K}}(M_{\textup {K}}) \subset M_{\textup {K}}.\) 
We next prove strict contractivity of the mapping \(\mathcal {T}_{\textup {K}}\)  in the energy norm for sufficiently small data. To this end, we take \(\phi ^{(1)}\)  and \(\phi ^{(2)}\)  in \(M_{\textup {K}}\) , and denote their difference by \(\overline{\phi }= \phi ^{(1)} -\phi ^{(2)} \) . Let then \(\psi ^{(1)}=\mathcal {T}_{\textup {K}}(\phi ^{(1)})\)  and \(\psi ^{(2)}=\mathcal {T}_{\textup {K}}(\phi ^{(2)})\) . The difference \(\overline{\psi }=\psi ^{(1)} -\psi ^{(2)} \in X^{\textup {K}}\)  solves the equation
\( \begin{aligned}(1-\kappa \phi _t^{(1)} )\overline{\psi }_{tt}-c^2 \Delta \overline{\psi }-b \Delta \overline{\psi }_t= \kappa \overline{\phi }_t \psi ^{(2)}_{tt}+\sigma \nabla \overline{\phi } \cdot \nabla \psi ^{(1)}_t+\sigma \nabla \phi ^{(2)} \cdot \nabla \overline{\psi }_t\end{aligned}\) 
with zero initial data. We test this equation with \(\overline{\psi }_t\)  and integrate over space and \((0,t)\)  to arrive at the identity
\( \begin{aligned}&\frac{1}{2}\Vert \sqrt{\alpha (t)}\,\overline{\psi }_t(t)\Vert _{L^2}^2+\frac{c^2}{2}\Vert \nabla \overline{\psi }(t)\Vert _{L^2}^2+b \Vert \nabla \overline{\psi }_t\Vert _{L^2(0, t; L^2)}^2 \\=&\,\begin{multlined}[t] -\frac{\kappa }{2} \int _0^t \int _{\Omega }\phi _{tt}^{(1)} \overline{\psi }^2_t \, \textup {d}x\textup {d}s+\kappa \int _0^t \int _{\Omega }\overline{\phi }_t\psi _{tt}^{(2)} \overline{\psi }_t \, \textup {d}x\textup {d}s\\+ \sigma \int _0^t \int _{\Omega } (\nabla \overline{\phi } \cdot \nabla \psi ^{(1)}_t+\nabla \phi ^{(2)} \cdot \nabla \overline{\psi }_t) \overline{\psi }_t \, \textup {d}x\textup {d}s, \end{multlined}\end{aligned}\) 
where \(\alpha (t)=1-\kappa \phi ^{(1)}_t(t)\) . To treat the \(\nabla \overline{\phi } \cdot \nabla \psi ^{(1)}_t\)  term on the right, we employ the following estimate:
\(\begin{aligned}\left|2\int _0^t \int _{\Omega } \nabla \overline{\phi } \cdot \nabla \psi ^{(1)}_t \overline{\psi }_t \, \textup {d}x\textup {d}s\right| \le &\, 2\Vert \nabla \psi ^{(1)}_t\Vert _{L^2(0,t ;L^\infty )} \Vert \nabla \overline{\phi }\Vert _{L^\infty (0,t; L^2)} \Vert \overline{\psi }_t\Vert _{L^2(0,t; L^2)}.\end{aligned}\) 
Similarly to (REF )–(REF ), to estimate the \(\nabla \phi ^{(2)} \cdot \nabla \overline{\psi }_t\)  term, we can rely on
\( \begin{aligned}\left|2\int _0^t \int _{\Omega } (\nabla \phi ^{(2)} \cdot \nabla \overline{\psi }_t) \overline{\psi }_t \, \textup {d}x\textup {d}s\right|=&\, \left| \int _0^t \int _{\Omega } (- ^{(2)}) \overline{\psi }_t \overline{\psi }_t \, \textup {d}x\textup {d}s\right|\\\le &\, \Vert ^{(2)}\Vert _{L^\infty (0,t; L^\infty )}\Vert \overline{\psi }_t\Vert _{L^2(0,t; L^2)}^2.\end{aligned}\) 
Additionally, we can bound the two \(\kappa \)  terms on the right-hand side of (REF ) as follows:
\(\begin{aligned}&\left|-\frac{1}{2}\kappa \int _0^t \int _{\Omega }\phi _{tt}^{(1)} \overline{\psi }^2_t \, \textup {d}x\textup {d}s+\kappa \int _0^t \int _{\Omega }\psi _{tt}^{(2)}\overline{\phi }_t \overline{\psi }_t \, \textup {d}x\textup {d}s\right| \\\le &\,\begin{multlined}[t] \frac{1}{2} |\kappa | \Vert \phi _{tt}^{(1)}\Vert _{L^\infty (0,t; L^\infty )}\Vert \overline{\psi }_t\Vert _{L^2(0,t; L^2)}^2 + |\kappa |\Vert \psi _{tt}^{(2)}\Vert _{L^2(0,t; L^\infty )} \Vert \overline{\phi }_t\Vert _{L^\infty (0,t; L^2)}\Vert \overline{\psi }_t\Vert _{L^2(0,t; L^2)}. \end{multlined}\end{aligned}\) 
Employing the above estimates in (REF ), using Young's inequality, and then taking the supremum over
\(t \in (0, \tau )\)  leads to
\( \begin{aligned}&\frac{1}{2} \underline{\alpha }\sup _{t\in (0,\tau )}\Vert \overline{\psi }_t(t)\Vert _{L^2}^2+\frac{c^2}{2}\sup _{t\in (0,\tau )}\Vert \nabla \overline{\psi }(t)\Vert _{L^2}^2+b \Vert \nabla \overline{\psi }_t\Vert ^2_{L^2(0,\tau ;L^2)} \\\le &\,\begin{multlined}[t] \frac{1}{2} |\kappa | \Vert \phi _{tt}^{(1)}\Vert _{L^\infty (L^\infty )}\Vert \overline{\psi }_t\Vert _{L^2(0,\tau ;L^2)}^2+\frac{1}{2} |\kappa \Vert \psi _{tt}^{(2)}\Vert _{L^2(L^\infty )}|\Vert \overline{\psi }_t\Vert _{L^2(0,\tau ;L^2)}^2 \\+ \frac{1}{2}|\kappa | \Vert \psi _{tt}^{(2)}\Vert _{L^2(L^\infty )}\Vert \overline{\phi }_t\Vert _{L^\infty (0,\tau ;L^2)}^2+\sigma \Vert \nabla \psi ^{(1)}_t\Vert _{L^2(L^\infty )} \Vert \nabla \overline{\phi }\Vert _{L^\infty (0,\tau ;L^2)}^2\\+\sigma \Vert \nabla \psi ^{(1)}_t\Vert _{L^2(L^\infty )}\Vert \overline{\psi }_t\Vert _{L^2(0,\tau ;L^2)}^2 +\sigma \Vert ^{(2)}\Vert _{L^\infty ( L^\infty )}\Vert \overline{\psi }_t\Vert ^2_{L^2(0,\tau ;L^2)}. \end{multlined}\end{aligned}\) 
Since the functions \(\phi ^{(1)}\)  and \(\phi ^{(2)}\)  belong to \(M_{\textup {K}}\)  and functions \(\psi ^{(1)}\)  and \(\psi ^{(2)}\)  solve the linear problem, we have the uniform bounds
\(\begin{aligned}\Vert \phi _{tt}^{(1)}\Vert _{L^\infty (L^\infty )} \le &\, C_{H^2, L^\infty }R_{\textup {K}}, \\\Vert ^{(2)}\Vert _{L^\infty ( L^\infty )} \le &\, C_{H^2, L^\infty }R_{\textup {K}},\\\Vert \psi _{tt}^{(2)}\Vert _{L^2(L^\infty )} \le &\, C_{H^2, L^\infty }\sqrt{T} C_{\textup {lin}, \textup {K}}(T,R_{\textup {K}}) (\Vert \psi _0\Vert _{H^4}+\Vert \psi _1\Vert _{H^3}), \\\Vert \nabla \psi ^{(1)}_t\Vert _{L^2(L^\infty )} \le & \, C_{H^2, L^\infty }\sqrt{T}\Vert \nabla \psi ^{(1)}_t\Vert _{L^\infty (H^2)} \\\le &\, \begin{multlined}[t]C_{H^2, L^\infty }\sqrt{T}\left((1+T+T^{3/2}C_{\textup {lin}, \textup {K}}(T,R_{\textup {K}}))^2 +C_{\textup {lin}, \textup {K}}(T,R_{\textup {K}})^2\right)^{1/2}\\\times (\Vert \psi _0\Vert _{H^4}+\Vert \psi _1\Vert _{H^3}), \end{multlined}\end{aligned}\) 
where we have estimated
\(\begin{aligned}\Vert \nabla \psi ^{(1)}_t\Vert _{L^\infty (H^2)}^2\le &\,\Vert \nabla \psi ^{(1)}_t\Vert _{L^\infty (L^2)}^2+\Vert \nabla ^{(1)}_t\Vert _{L^\infty (L^2)}^2\\\le &\, (\Vert \nabla \psi _1\Vert _{L^2}+T\Vert \nabla \psi _2\Vert _{L^2}+T^{3/2}\Vert \nabla \psi ^{(1)}_{ttt}\Vert _{L^2 (L^2)})^2+\Vert \nabla ^{(1)}_t\Vert _{L^\infty (L^2)}^2.\end{aligned}\) 
Therefore, applying Gronwall's inequality to (REF ) results in the bound
\(\begin{aligned}\Vert \overline{\psi }\Vert ^2_{\textup {E}} \le &\, \begin{multlined}[t] C_1(1+T^{3/2}) \, \sqrt{T}\, (\Vert \psi _0\Vert _{H^4}+\Vert \psi _1\Vert _{H^3}) \\ \times \exp (C_2 (R_{\textup {K}}+(1+T^{3/2})\sqrt{T}(\Vert \psi _0\Vert _{H^4}+\Vert \psi _1\Vert _{H^3}))\Vert \overline{\phi }\Vert ^2_{\textup {E}}, \end{multlined}\end{aligned}\) 
where the constants \(C_1\) , \(C_2>0\)  do not depend on \(b\)  nor \(T\) . By sufficiently reducing the final time so that
\(\begin{aligned}\begin{multlined}[t]C_1(1+T^{3/2}) \, \sqrt{T}\, (\Vert \psi _0\Vert _{H^4}+\Vert \psi _1\Vert _{H^3})\\ \times \exp (C_2 (R_{\textup {K}}+(1+T^{3/2})\sqrt{T}(\Vert \psi _0\Vert _{H^4}+\Vert \psi _1\Vert _{H^3})) <1,\end{multlined}\end{aligned}\) 
we can guarantee that
\(\Vert \psi ^{(1)}-\psi ^{(2)}\Vert _{\textup {E}} \le r \Vert \phi ^{(1)}-\phi ^{(2)}\Vert _{\textup {E}} \ \text{ for some } r \in (0,1). \) 
We note that it would be possible to impose smallness of data in the \(H^4 \times H^3\)  norm instead of reducing the final time, analogously to the proof of well-posedness for the Westervelt equation.
We can argue similarly to the proof of Theorem REF  that \((M_{\textup {K}}, d_{\textup {E}})\)  is a closed subset of a complete space with the metric induced by the energy norm \(d_{\textup {E}}(x,y)=\Vert x-y\Vert _{\textup {E}}\) . An application of Banach's Fixed-point theorem to \(\mathcal {T}\)  thus concludes the proof.

The inviscid limit of the Kuznetsov equation
Having obtained the uniform bounds with respect to \(b\)  in the \(X^{\textup {K}}\)  norm, we are now ready to derive the convergence rate for solutions of the Kuznetsov equation as \(b \rightarrow 0^+\) .
Theorem 7.1 
Under the assumptions of Theorem REF , the family of solutions \(\lbrace \psi ^{(b)}\rbrace _{b>0}\)  of the Kuznetsov equation converges in the energy norm to a solution \(\psi \)  of the inviscid Kuznetsov equation at a linear rate. In other words,
\( \Vert \psi ^{(b)}-\psi \Vert _{\textup {E}}\lesssim b \ \mbox{ as } b\rightarrow 0.\) 

The proof follows along the lines of the proof of contractivity in the previous theorem. Let \(b, b^{\prime } \in (0, \overline{b})\) . Let \(\psi ^{(b)}\)  and \(\psi ^{(b^{\prime })}\)  be the solutions of the Kuznetsov equation with the sound diffusivity \(b\)  and \(b^{\prime }\) , respectively, and with homogeneous Dirichlet data, where we assume that initial conditions and final time are chosen according to the well-posedness result of Theorem REF .
The difference \(\overline{\psi }=\psi ^{(b)}-\psi ^{(b^{\prime })}\)  then solves the equation
\( \begin{aligned}&(1-\kappa \psi _t^{(b)} )\overline{\psi }_{tt}-c^2 \Delta \overline{\psi }-b \Delta \overline{\psi }_t\\=&\, \begin{multlined}[t] \kappa \overline{\psi }_t \psi ^{(b^{\prime })} _{tt}+\sigma \nabla \overline{\psi }\cdot \nabla \psi ^{(b)}_t+\sigma \nabla \psi ^{(b^{\prime })} \cdot \nabla \overline{\psi }_t+(b-b^{\prime })\Delta \psi _t^{(b^{\prime })}, \end{multlined}\end{aligned}\) 
supplemented by zero initial conditions. Testing with \(\overline{\psi }_t\)  and integrating with respect to space and time yields
\( \begin{aligned}&\frac{1}{2}\Vert \sqrt{\alpha (t)}\overline{\psi }_t(t)\Vert _{L^2}^2+\frac{c^2}{2}\Vert \nabla \overline{\psi }(t)\Vert _{L^2}^2+b \Vert \nabla \overline{\psi }_t\Vert _{L^2(0, t; L^2)}^2 \\=&\,\begin{multlined}[t] -\frac{\kappa }{2} \int _0^t \int _{\Omega }\psi _{tt}^{(b)} \overline{\psi }^2_t \, \textup {d}x\textup {d}s+\kappa \int _0^t \int _{\Omega }\psi _{tt}^{(b^{\prime })} \overline{\psi }^2_t \, \textup {d}x\textup {d}s\\+ \sigma \int _0^t \int _{\Omega } (\nabla \overline{\psi }\cdot \nabla \psi ^{(b)}_t+\nabla \psi ^{(b^{\prime })} \cdot \nabla \overline{\psi }_t) \overline{\psi }_t \, \textup {d}x\textup {d}s\\+ (b-b^{\prime })\int _0^t \int _{\Omega }_t^{(b^{\prime })} \overline{\psi }_t \, \textup {d}x\textup {d}s, \end{multlined}\end{aligned}\) 
where \(\alpha (t)=1-\kappa \psi ^{(b)}_t(t)\) . We can rely on the following estimate:
\(\begin{aligned}\left|\int _0^t \int _{\Omega } \nabla \overline{\psi }\cdot \nabla \psi ^{(b)}_t \overline{\psi }_t \, \textup {d}x\textup {d}s\right| \le \Vert \nabla \psi ^{(b)}_t\Vert _{L^\infty (0,t ;L^\infty )} \Vert \nabla \overline{\psi }\Vert _{L^2(0,t; L^2)} \Vert \overline{\psi }_t\Vert _{L^2(0,t; L^2)}.\end{aligned}\) 
Similarly to (REF ), we also have the bound
\(\begin{aligned}\left|\int _0^t \int _{\Omega } (\nabla \psi ^{(b^{\prime })} \cdot \nabla \overline{\psi }_t) \overline{\psi }_t \, \textup {d}x\textup {d}s\right|=&\, \left|\frac{1}{2} \int _0^t \int _{\Omega } (- ^{(b^{\prime })}) \overline{\psi }_t \overline{\psi }_t \, \textup {d}x\textup {d}s\right|\\\le &\, \frac{1}{2} \Vert ^{(b^{\prime })}\Vert _{L^\infty (0,t; L^\infty )}\Vert \overline{\psi }_t\Vert _{L^2(0,t; L^2)}^2.\end{aligned}\) 
The two \(\kappa \)  terms on the right-hand side of (REF ) can be estimated as follows:
\(\begin{aligned}&\left|-\frac{1}{2}\kappa \int _0^t \int _{\Omega }\psi _{tt}^{(b)} \overline{\psi }^2_t \, \textup {d}x\textup {d}s+\kappa \int _0^t \int _{\Omega }\psi _{tt}^{(b^{\prime })} \overline{\psi }^2_t \, \textup {d}x\textup {d}s\right| \\\le &\, |\kappa | \left(\frac{1}{2} \Vert \psi _{tt}^{(b)}\Vert _{L^\infty (0,t; L^\infty )}+ \Vert \psi _{tt}^{(b^{\prime })}\Vert _{L^\infty (0,t; L^\infty )}\right)\Vert \overline{\psi }_t\Vert _{L^2(0,t; L^2)}^2.\end{aligned}\) 
Thus, employing the above estimates in (REF ), Young's inequality, and taking the supremum over \(t \in (0, \tau )\)  leads to
\( \begin{aligned}&\frac{1}{2} \underline{\alpha }\sup _{t\in (0,\tau )}\Vert \overline{\psi }_t(t)\Vert _{L^2}^2+\frac{c^2}{2}\sup _{t\in (0,\tau )}\Vert \nabla \overline{\psi }(t)\Vert _{L^2}^2+b \Vert \nabla \overline{\psi }_t\Vert ^2_{L^2(0,\tau ; L^2)} \\\le &\,\begin{multlined}[t] |\kappa | \left(\frac{1}{2}\Vert \psi _{tt}^{(b)}\Vert _{L^\infty (L^\infty )}+\Vert \psi _{tt}^{(b^{\prime })}\Vert _{L^\infty ( L^\infty )}\right)\Vert \overline{\psi }_t\Vert _{L^2(L^2)}^2 \\+ \frac{1}{2}|\sigma |\Vert \nabla \psi ^{(b)}_t\Vert ^2_{L^\infty (L^\infty )} \Vert \nabla \overline{\psi }\Vert _{L^2( L^2)}^2+ \frac{1}{2}|\sigma |\Vert \overline{\psi }_t\Vert ^2_{L^2( L^2)}\\+\frac{1}{2}|\sigma |\Vert ^{(b^{\prime })}\Vert _{L^\infty (L^\infty )}\Vert \overline{\psi }_t\Vert _{L^2( L^2)}^2 + \frac{1}{2} (b-b^{\prime })^2 \Vert _t^{(b^{\prime })}\Vert _{L^2( L^2)}^2+\frac{1}{2}\Vert \overline{\psi }_t\Vert _{L^2( L^2)}^2. \end{multlined}\end{aligned}\) 
By virtue of Theorem REF , we have the uniform bound
\(\begin{aligned}\begin{multlined}[t] \Vert \psi _{tt}^{(b)}\Vert _{L^\infty (L^\infty )}+\Vert \psi _{tt}^{(b^{\prime })}\Vert _{L^\infty (L^\infty )}+\Vert \nabla \psi ^{(b)}_t\Vert ^2_{L^\infty (L^\infty )} \\+\Vert ^{(b^{\prime })}\Vert _{L^\infty (L^\infty )} \lesssim R_{\textup {K}} + R_{\textup {K}}^2, \end{multlined}\end{aligned}\) 
and \(\Vert _t^{(b^{\prime })}\Vert _{L^2 (L^2)}^2 \lesssim T R_{\textup {K}}^2\) . Therefore applying Gronwall's inequality results in
\(\Vert \overline{\psi }\Vert _{\textup {E}}=\Vert \psi ^{(b)}-\psi ^{(b^{\prime })}\Vert _{\textup {E}} \lesssim |b-b^{\prime }|.\) 
Finally, we note that the estimate remains valid for \(b^{\prime }=0\)  and thus (REF ) holds, as claimed.

Conclusion
In this work, we have performed a convergence study of parabolic approximations to quasilinear wave equations of second order as the damping parameter \(b\)  tends to zero. Our research has been motivated by two classical nonlinear acoustics models, the Westervelt and Kuznetsov equations. As the analysis showed, the solutions of these equations converge in the energy norm with a linear rate to the solutions of the respective inviscid equations (with \(b=0\) ), provided that the initial data are smooth and small and the final time short enough. For a short enough final time, the smallness condition on the initial data can be imposed in a lower-order space than the one needed to derive higher-energy bounds. In particular, the smallness condition on the data only arises from ensuring that the non-degeneracy assumption on \(p\)  and \(\psi _t\)  holds in the Westervelt and Kuznetsov equations, respectively.

Acknowledgment
The work of the first author was supported by the Austrian Science Fund fwf under the grants P30054 and DOC 78.
